Selected for a Viewpoint in Physics PHYSICAL REVIEW X 3, 021002 (2013) Fermi Surface of the Most Dilute Superconductor Xiao Lin,1,2 Zengwei Zhu,1 Benoıˆt Fauque´,1 and Kamran Behnia1 1LPEM (UPMC-CNRS), Ecole Supe´rieure de Physique et de Chimie Industrielles, 75005 Paris, France 2Department of Physics, Zhejiang University, Hangzhou, 310027, China (Received 5 December 2012; published 15 April 2013) The origin of superconductivity in bulk SrTiO3 is a mystery since the nonmonotonous variation of the critical transition with carrier concentration defies the expectations of the crudest version of the BCS theory. Here, employing the Nernst effect, an extremely sensitive probe of tiny bulk Fermi surfaces, we show that, down to concentrations as low as 5:5  1017 cmÀ3, the system has both a sharp Fermi surface and a superconducting ground state. The most dilute superconductor currently known therefore has a metallic normal state with a Fermi energy as little as 1.1 meV on top of a band gap as large as 3 eV. The occurrence of a superconducting instability in an extremely small, single-component, and barely anisotropic Fermi surface implies strong constraints for the identification of the pairing mechanism. DOI: 10.1103/PhysRevX.3.021002 Subject Areas: Semiconductor Physics, Strongly Correlated Materials, Superconductivity I. INTRODUCTION At still higher doping levels, the system becomes a superconductor when it is n doped by one of the SrTiO3 is a large-gap transparent insulator, which, upon three possible routes: substituting titanium with niobium the introduction of n-type carriers, undergoes a supercon- [11–13], strontium with lanthanum [14], or removing ducting transition below 1 K. Discovered in 1964 [1], it is the first member of a loose family of ‘‘semiconducting oxygen [11]. Intriguingly, the superconducting ground superconductors’’ [2], which now includes column-IV state is restricted to a limited doping window [11–14]. elements [3]. During the last decade, attention has been Thanks to the remarkably high mobility of electrons, another consequence of the large dielectric constant [10], focused on the interface between SrTiO3 and other insu- lators [4] or a vacuum [5], which is a two-dimensional quantum oscillations are observable and have been metal with a superconducting ground state [6,7]. reported both in the bulk [15,16] and in the two- According to Mott, in a doped semiconductor, when dimensional [17–20] samples. the average distance between the dopants (d ¼ nÀ1=3) The main subject of the present work is to yield a becomes a sizeable fraction of the effective Bohr radius quantitative description of the emerging Fermi surface à and to clarify its relationship with the superconducting aB, a metal-insulator transition occurs. Quantitatively, this Mott criterion for the critical concentration is expressed as ground state. Our work addresses two hitherto unanswered 1=3 à questions. (i) Is there a threshold in carrier concentration nc a ¼ 0:26 and has been observed to hold in a wide B for the emergence of superconductivity? (ii) Does the range of semiconductors [8]. normal state of such a low-density superconductor have a Insulating SrTiO3 is dubbed a quantum paraelectric. well-defined Fermi surface, or is it an impurity-band At low temperature, its static dielectric constant becomes metal? We provide answers to these questions by a study 4 orders of magnitude larger than a vacuum [9]. Since the of the low-temperature Nernst effect (a very sensitive effective Bohr radius is proportional to the dielectric probe of tiny Fermi surfaces [21–23]) in both oxygen- constant, this large dielectric constant implies a very long reduced and Nb-doped SrTiO3 across a wide (i.e., Bohr radius. Therefore, following the Mott criterion, the critical density for a metal-insulator transition is expected 3-orders-of-magnitude) window of carrier density. to be much lower than in ordinary semiconductors, and, We find that superconductivity persists down to a carrier concentration that is significantly lower than what indeed, doped SrTiO3 displays a finite zero-temperature was previously believed. This result firmly establishes conductivity down to carrier concentrations as low as n SrTiO n ¼ 8  1015 cmÀ3 [10], orders of magnitude lower than -doped 3 as the most dilute known superconductor 5:5  1017 cmÀ3 the threshold of metallicity in silicon (3:5  1018 cmÀ3)or with a carrier density as low as , which 105 in germanium (3:5  1017 cmÀ3)[8]. corresponds to the removal of one oxygen atom out of . At this carrier concentration, giant Nernst quantum oscillations with a single frequency are observed. The superconductivity persists even in the presence of a single, barely filled, and almost isotropic band. We will argue that Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- this context is radically different from what one finds in bution of this work must maintain attribution to the author(s) and conventional phonon-mediated BCS superconductors. the published article’s title, journal citation, and DOI. Thus, more than four decades after its discovery, SrTiO3 2160-3308=13=3(2)=021002(8) 021002-1 Published by the American Physical Society LIN et al. PHYS. REV. X 3, 021002 (2013) emerges from this study as a candidate for unconventional room-temperature resistivity were found to be compatible superconductivity. with early reports by Fredrikse and co-workers [24,25] and recent work by Spinelli and co-workers [10]. In order to II. EXPERIMENTAL obtain Ohmic contacts, gold was evaporated on the samples and heated up to 550 C. The typical contact The study was carried out on bulk commercial SrTiO3 resistance was a few Ohms at room temperature and well single crystals. Oxygen-deficient samples were obtained below 1 Ohm at low temperature. A one-heater–two- by annealing nominally stochiometric samples. The thermometers setup was used to measure all transverse Hall and longitudinal resistivity of the samples were and longitudinal electric and thermoelectric coefficients measured, and the carrier dependences of mobility and (resistivity, Hall, Seebeck, and Nernst). A sketch of the setup is shown in the upper panel of Fig. 1.Atlow temperatures, a noise level of 1 nV was achieved. The accuracy of the thermal gradient across the sample was checked by retrieving the Wiedemann-Franz relation between thermal and electric conductance in each sample. Details on the sample preparation and characterization are given in the Supplemental Material [26]. III. RESULTS AND DISCUSSION Figure 1 presents the variation of the Nernst signal (Sxy ¼ Ey= rxT) as a function of the inverse of the FIG. 1. Top: The experimental setup for measuring Nernst and FIG. 2. Nernst signal as a function of the inverse of the Seebeck coefficients of bulk chemically doped SrTiO3.Bottom:As magnetic field for two carrier concentrations. The quantum the carrier density is reduced in SrTiO3, the Nernst signal shows oscillations display a complex structure with multiple periodici- oscillations with larger amplitude and longer periodicity. In each ties. The inset shows a sketch of the position of Fermi energy panel, the vertical red bar represents a constant scale of 0:7 V=K. relative to the two lowest bands at the À point. 021002-2 FERMI SURFACE OF THE MOST DILUTE SUPERCONDUCTOR PHYS. REV. X 3, 021002 (2013) magnetic field for seven samples with different carrier other hand, the contribution of normal quasiparticles to the concentrations kept at the same temperature (0.5 K). As Nernst signal is proportional to the ratio of their mobility is seen in the figure, quantum oscillations are barely de- to their Fermi energy [28]. In a dirty high-density two- tectable in the sample with the highest carrier density. As n dimensional superconductor with low electron mobility decreases, the amplitude of oscillations grows and their and large Fermi energy, the quasiparticle contribution is frequency shrinks. Giant oscillations of the Nernst effect small, and the superconducting Nernst signal is detectable with the approach of the quantum limit were previously in a large temperature window above Tc [29]. In slightly observed in semimetallic bismuth [21] and graphite [22], doped bulk SrTiO3, in contrast, the electron mobility is as well as doped Bi2Se3 [23]. A property that these three large, the Fermi energy is small, and the Nernst signal is systems share with lightly doped SrTiO3 is that their Fermi therefore dominated by the quasiparticle contribution. (See surface is an extremely small portion of the Brillouin zone. the Supplemental Material for a comparison of the orders A 10 T magnetic field truncates such a tiny Fermi surface of magnitude of the quasiparticle and superconducting into a few Landau tubes. In such a context, each time a contributions to the Nernst signal [26]). squeezed Landau tube leaves the Fermi surface, the Nernst As is seen in Fig. 1, in the intermediate doping range, signal peaks. The Nernst quantum oscillations are concom- the oscillations display a complex structure, and several itant with Shubnikov–de Haas oscillations. For the lowest frequencies are detectable. For samples with larger carrier Landau indexes, however, while the oscillating component densities (n 1:05  1018 cmÀ3), the spectrum of oscil- of resistivity is a small fraction of the overall signal, the lations is indicative of the presence of more than one oscillating part of the Nernst coefficient dominates the component of the Fermi surface. For the lowest doping monotonous background [22], making the analysis of level, the structure becomes remarkably simpler. the Nernst data straightforward. Figure 2 displays detailed data at different temperatures At low magnetic fields, the monotonous Nernst signal is for two low-density samples. Quantum oscillations show a expected to be affected by fluctuating superconductivity. complex structure.